Application of Quilghini transformation method to exact solutions of Stefan's models for thermal dissociation hydrate with density jump

doi:10.11949/0438-1157.20240509

Abstract

Abstract:

Stefan's models for thermal decomposition of natural gas hydrate with density jump is established by using the principle of mass conservation. The general form of Rankine-Hugoniot jump relation was derive in detail, and the Stefan coupling condition in cubic form with respect to the velocity of interface was obtained, but the cubic term could be ignored within the errors of engineering requirements. It is the introduction of velocity terms in the Stefan model that makes it difficult to obtain an exact solution to the model. The mass as a space variable was introduced by the Quilghini transformation, the velocity term was vanished, and become the classical Stefan-like's models. Followed by the inverse Quilghini transformation, the accurate analytical solutions of the Stefan's models with density jump could be successfully gained. Through Matlab programming, a concrete example was taken, laws on the solutions of transcendental equation, temperature distribution and dissociation interface were studied, the influences of injection temperature and density jump on solutions of transcendental equation, penetration depth, penetration time and hydrate displacement were probed. The errors caused by density jump were within the permissible ranges of engineering.

Key words: Quilghini transformation, density jump, Stefan's model, Rankine-Hugoniot jump relation, natural gas hydrate

摘要：

利用质量守恒原理建立了考虑密度跃变的天然气水合物热分解Stefan模型，通过对有弱解的均匀散度方程分部积分详细推导一般形式的Rankine-Hugoniot跳跃关系，获得了关于界面速度项三次方形式的Stefan耦合条件，但在工程需要误差范围内三次方可以忽略。在Stefan模型中引入速度项使得模型的精确解难以得到。Quilghini变换将质量转化为空间变量，Stefan模型的速度项消失，成为经典的类Stefan模型，再经Quilghini逆变换得到了密度跃变Stefan模型的精确解析解。通过Matlab编程，结合实例研究了超越方程解、温度分布和分解界面的规律性，探索注入温度和密度跃变对超越方程解、穿透深度、穿透时间和水合物位移的影响，密度跃变引起的误差在工程允许范围内。

关键词: Quilghini变换, 密度跃变, Stefan模型, Rankine-Hugoniot跳跃关系, 天然气水合物

CLC Number:

TQ 511.1

Mingchuan LI, Shuanshi FAN, Fuhai XU, Huidong LU, Xiaojun LI. Application of Quilghini transformation method to exact solutions of Stefan's models for thermal dissociation hydrate with density jump[J]. CIESC Journal, 2024, 75(10): 3763-3774.

李明川, 樊栓狮, 徐赋海, 卢惠东, 李晓军. Quilghini变换法在求解密度跃变的水合物热分解Stefan模型精确解中的应用[J]. 化工学报, 2024, 75(10): 3763-3774.

Figures/Tables 12

Fig.1 Dissociation interface S and an across curve Σ

Fig.2 Calculation flowchart

Table 1 Thermo-physical parameter of slab one-dimensional hydrate reservoir

初始压力/MPa

初始温度/℃

密度/(kg·m^-3)

液相水合物分解区

热扩散系数/(m²·s^-1)

固相水合物区

热扩散系数/(m²·s^-1)

液相水合物分解区

热传导率/(W·m^-1·K^-1)

固相水合物区热传导率/(W·m^-1·K^-1)

Fig.3 Transcendental equation f(λ) vsλ

Fig.4 Transcendental equation f(λ) vsλ with effects of Tinj and ρH

Fig.5 Temperature vs distance, δxs(t) and xdvs time

Fig.6 Temperature vs distancewith effects of Tinj and ρH

Fig.7 Temperature vs time,δxs(t) and tdvss(t)

Fig.8 Temperature vs time with effects of Tinj and ρH

Fig.9 s(t) vs time with effects of Tinj and ρH

Table 1 Variable fitting equations with effects of Tinj，ρH

序号	变量	拟合函数(影响因素)		ρ_H引起的误差/%	其他	来源
序号	变量	T_inj	ρ_H	ρ_H引起的误差/%	其他	来源
1	λ	λ=-2.713 $T i n j - 0.1068$ +4.469	λ=0.2857exp(-0.002816ρ_H) + 2.902exp(-4.352×10^-5ρ_H)	0.532999463		图4
2	x_d	x_d=0.4952 $T i n j - 0.1079$ +0.9248	x_d=0.6255exp(0.0007971ρ_H)-39.31exp(-0.009085ρ_H)	1.897220537	x_d=0.3797t^0.4847+0.1226	图5, 图6
3	t_d	t_d=-2.399×10^-7 $T i n j 3$ +0.0001528 $T i n j 2$ -0.03428T_inj+5.262	t_d=0.03634 $ρ H 0.682$	2.977667494	t_d=-0.01667[s(t)]³+0.8089[s(t)]²-9.323s(t)+39.67	图7,图8
4	s(t)	s(t)=-8.573 $T i n j - 0.1068$ +14.12	s(t)= -5.773 $ρ H 0.6051$ +17.61	0.776644191		图9
5	t	t=21.07 $T i n j - 0.2943$ +7.232	t=-8.963×10^-6 $ρ H 2$ +0.003279ρ_H+10.42	0.535378675		图9

Table 1 Variable fitting equations with effects of Tinj，ρH

序号	变量	拟合函数(影响因素)		ρ_H引起的误差/%	其他	来源
序号	变量	T_inj	ρ_H	ρ_H引起的误差/%	其他	来源
1	λ	λ=-2.713 $T i n j - 0.1068$ +4.469	λ=0.2857exp(-0.002816ρ_H) + 2.902exp(-4.352×10^-5ρ_H)	0.532999463		图4
2	x_d	x_d=0.4952 $T i n j - 0.1079$ +0.9248	x_d=0.6255exp(0.0007971ρ_H)-39.31exp(-0.009085ρ_H)	1.897220537	x_d=0.3797t^0.4847+0.1226	图5, 图6
3	t_d	t_d=-2.399×10^-7 $T i n j 3$ +0.0001528 $T i n j 2$ -0.03428T_inj+5.262	t_d=0.03634 $ρ H 0.682$	2.977667494	t_d=-0.01667[s(t)]³+0.8089[s(t)]²-9.323s(t)+39.67	图7,图8
4	s(t)	s(t)=-8.573 $T i n j - 0.1068$ +14.12	s(t)= -5.773 $ρ H 0.6051$ +17.61	0.776644191		图9
5	t	t=21.07 $T i n j - 0.2943$ +7.232	t=-8.963×10^-6 $ρ H 2$ +0.003279ρ_H+10.42	0.535378675		图9

Table 2 Fitting equation of δxs（t）

序号	变量	其他拟合函数		来源
1	δx_s₍_t₎	δx_s₍_t₎=-0.07689exp(-0.1695Δt)-0.03523exp(-0.01509Δt)	δx_s₍_t₎=0.0009647[s(t)]³-0.02562[s(t)]²+0.1798s(t)-0.4583	图5,图7

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